Proceedings of the 2008 International Conference on Electrical Machines Paper ill 1365

Torque and Current Ripple Analytical Investigation in Space-Vector PWM Inverter Fed Induction Motor Drive under DC-Bus Voltage Pulsation

Jiri Klima *,Miroslav Chomat**,Ludek Schreier ** *Dep. of Electrical Eng. and Automation., Faculty of Engineering

University of Life Science in Prague 16521 Prague6-Suchdol,Czech Republic

Tel: (+420224384228, fax: (+420)224383203 e-mail: klima@t(czu.cz

** Department of Electrical Machines,Institute of Thermomechanics, Douejskova 5,Prague 8,18000

Te.(+420266052394) e-mail: chomat@it.cas.cz

Abstract- Analytical investigation of the current and electromagnetic torque ripples in the three-phase voltage source inverter fed induction motor drive under DC-link voltage pulsation is presented in tbis paper. The analytical expressions for the voltage and current space-vectors as a function of the DC-link voltage pulsation are derived. The analytical results are verified by the experiments.

I. INTRODUCTION

The use of voltage source inverters (VSI) with DC links for feeding of induction motors is getting more and more common. But voltage unbalance orland sag conditions generated by the line supply can cause the input rectifier stage to enter the single-phase operation with the harmonics of the twice supply frequency. Under input voltage unbalance, or sag conditions, the presence of the second line frequency (2tj) harmonic voltage components in the dc bus voltage generates low-frequency harmonic in the output inverter voltage which can have series performance consequences in adjustable-speed drive. An increase in electric losses, excessive rise of the motor temperature, appearance of the torque pulsation, and noise problems are just some ofthe possible problems [I]. SO, the analysis of the DC-link pulsation and its influence on the motor currents and an electromagnetic torque is of great importance [3]. [4]. Similar investigation was presented in [2]. but in this work the inverter output voltage is approximated by only the fundamental harmonic. In this paper, analytical model and closed-form analysis of the DC-link pulsation effects on the current and torque ripples are presented. The analytical closed-form expressions are based on the mixed p-z approach [8]. The analytical results are verified by the experiments.

II. ANALYTICAL ANALYSIS

As we investigate the periodic discrete signals we express time in per units as

978-1-4244-1736-0/08/$25.00 ©2008 IEEE

t = (n+E)T = (n+E)To /6 , (1)

where T is the interval of periodicity that is called a sector interval, n is the number of a sector, E is a per unit time inside a sector, 0 ~ E ~ I and To is the fundamental output frequency of the inverter.

In the case of the dc link voltage ripple we can express the voltage space-vector of the motor in the n-th sector with constant and pulsation parts as follows:

2 jn!!: V(n, e) ="3[ Vdc + L1Vdc cos [2li.\ (n +e)TlJe 3 (2)

= Vc(n, e) + Vp(n,e)

where the two parts are given as follows: The voltage space vector from the pulsation ofthe DC-link voltage can be calculated as follows:

2 jn!£ Vp(n,E)="3L1Vdee 3 cos(2111(n+E)T)

= .!.. L1Vdeejn~ e j2ll.\ (n+e)T + 3

1 jn!£. -L1V e 3 e-J2ll.\(n+e)T = V (n E)+ V (n E) (3) 3 de pP' pN'

2 jn!£ Ve(n,E)=-Vdee 3

3 Vpp(n, E) and V pN(n, E) are, respectively, the positive and negative components of the pulsation part of the dc bus voltage.

j1t.ak Lf(e,k).e 3 is the modulation function given by the k

switching instants in k-the pulse. This function contains both time and phase vector dependency.

Proceedings of the 2008 International Conference on Electrical Machines

f(E,k) = 1,for .. .EkA ~ E ~ EkB , else f(E,k) = 0

(3a) where: EkA is start point setting per unit time ofk-the pulse

EkB is end point setting per unit time ofk-the pulse

duty ratio switching time:

~k =EkB -EkB

From Fig.1a we can see the trajectories of the both voltage space vectors Vc(n, £. ) and Vp(n, £. ) in a - ~ complex plane for

The amplitude of the dc-link pulsation AV"" = 0.1

The real part of the pulsation space vector Re(vp(n, £.» which is also waveform in the phase "a" is shown in Fig. 1 b. By separating the voltage space vector we can investigate the influence of the both part on the currents and/or torque pulsation performance of the induction motor.

b)

Fig.l a)Trajectories of voltage space vector Vc(n, £.) and Vp(n, £.) in a-~ complex plane)Wavefonn of real part of voltage space vector Vp(n,£.).

III. VOLTAGE SPACE VECTORS

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Using the mixed p-z approach as shown in [9], we can derive the closed-form analytical expressions for the stator and/or rotor currents and also for the electromagnetic torque. Using (2) with the help of the modified Z transform of exponential functions we can find the Laplace transform of the stator voltage vector as follows:

V;(P)=~V<k: 1 ff'T ." L(e-PTs.. -e-PTe,.,,)+ 3 P_PT ')k

t;" -e

~,lV<k: 1. ff'T" L(e-'fI1.A(P-2j CQ,) _e-Te,.,,(P-2jCQ,»+ 3 p-2Jcq _pT j('3+2<qT) k

t;" -e

~,lV 1 ff'T L(e-'fI1.A(P+2jCQ,) _e-Te,.,,(P+2jCQ,»= 3 <k: p+2jcq _pT j(~-2cqT) k

t;" -e Yo(P)+ V;p(p-2jcq)+ V;N(P+ 2jcq)

(4) Equation (3) is the Laplace transform of the stator voltage vector, respecting the influence of the dc-link voltage ripple. Again, as in the time domain it contains three parts. The first term is the Laplace transform of the stator voltage vector with constant value in the dc link voltage. The last two terms are the Laplace transform of the positive and the negative ripple components of the dc link voltage. When we know the Laplace transform of the stator voltage vectors we can derive the Laplace transform of the motor current vectors. In order to calculate the motor current space vectors, it is convenient to carry out the analysis in the stator reference frame.

IV. PULSATION OF THE MOTOR CURRENTS

From the motor equations in the stator reference frame we can derive the Laplace transform of the stator and rotor currents, respectively as follows

(5)

In the foregoing equations, Rs is the stator resistance, RR the rotor resistance, Ls the stator self-inductance, LR the rotor self-inductance, Lm is the mutual inductance, and OJ

the rotor electrical angular velocity

(6)

Proceedings of the 2008 International Conference on Electrical Machines

and Pl,2 in (5) are the roots of the characteristic equation: B(p )=0, that are given as follows:

Pl,2 -(ks + kR - jO'co) ±

20'

(ks + kR - jO'CO)2 + jcoks - kskR 20' 0'

(7)

The inverse Laplace transform of (5) cannot be solved directly using the residue theorem, as (5) contains infinite numbers of poles given by the following equations (see eq, (4)):

,It

e~ _epT =0 (8)

j(~-2"'T) e 3 -eP T =0,

The solution of the time dependency of the motor current vectors can be found in the closed form as presented in [8], If we use the Heasivide theorem

The solution contains both the steady-state and transient components, As our attention is focused on the steady-state solution, applying superposition, the stator and rotor currents will have the closed-form solution shown in (9) , forn~,

.It( +1) 2 .It( I) Iy(n,e) = 1\ (O)elJ n + L 11y~)elJ n+ ePtTE +

k=1

j.!!(n+l) e l2jIDt)e 3 e2jCl\T(n+E) +

± e y~ _ 2jIDt)ej(~+2cq,T)(n+l) ePtTE + k=1 (9)

13 (2' ) ~(n+l) -2jCl\T(n+E) y - JIDt e e +

± 13 y~ + 2jIDt)ej(~-2cq,T)(n+l) ePtTE = k=1

lyO(n,e)+ Iyp(n,e)+ Iyn(n,e)

where subscripts in (9) mean: y is S (for the stator values) or y is R (for the rotor values), o are parts of (9) that are not dependent on angular

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frequency ~ ,p means a positive sequence terms

(depending on ~) and n means a negative sequence terms

(depending on -~). Similarly to (4) the overall steady-state stator/rotor currents contain three parts (terms arising from constant, positive and negative bus voltages, respectively).

Fig.2 Time depen<lency of the phase current pulsation part (top trace)

and overall phase currents (bottom trace) with 5% dc-bus voltage pulsation

From Fig.2 one can see the pulsation part (Spins) of the phase currents (top trace) and the overall phase currents Is for 5% of the DC-bus voltage pulsation . These phase currents were calculated from (9) by the inverse transformation (from space vectors to phase domain).

V. ELECTROMAGNETIC TORQUE

The electromagnetic torque can be determined again in closed-form as follows:

Where" means complex conjugate value

If we substitute analytical equations for the stator/rotor currents (9) into (10) we get the solution for the electromagnetic torque in closed form.

Ti (n,E) = t Lm ~ Re{j.Iso * (n, e).IRO (n,e) } +

t Lm ~ Re{j(lsp" (n,e) + Iso" (n,e))IRo(n, e)} +

tLm ~Re{jIso*(n,e)(lRP(n,e)+ IRn(n,E»} + (11)

+t Lm ~ Re{j(lsp" (n,e) + Iso" (n,e))(IRp(n,e) + IRn (n,e))}

Proceedings of the 2008 International Conference on Electrical Machines

This expression shows that, under unbalance input voltage conditions, the electromagnetic torque contains the following terms: TiO(n,E) is the electromagnetic torque without dc bus ripple

component. It contains an average dc term and sixth-harmonic pulsations caused by the inverter switching frequency. This torque component along with the overall torque waveform is shown in Fig.3. Til (n,E) andTi2 (n,E) represent the products of the complex conjugate stator currents form the constant (ripple) dc bus and the rotor currents from the ripple (constant) dc bus voltage. Both pulsation torque components and also their sum are shown in Fig.4.As can be seen in spite of high frequency pulsation separate components Tipulsl and Tipuls 2 their sum does not contain this high frequency pulsation, and it may be very closely approximate by the function

Tipuls 1 +Tipuls 2 = 0,078 sin (20)0 t) (12)

Tdn,E) is the electromagnetic torque component from both stator and rotor dc link pulsation parts. I has negligible value and we may put Tdn,E)=O. From that expression we can estimate the influence of the

torque pUlsation given from the dc-voltage ripple. As the voltage space vectors contain two parts, namely positive and negative components (see (3)) we obtain also two additional torque parts.

Torque ripple with low frequency may become an important issue in induction motor drives .It cause mechanical oscillations, which are particularly dangerous in resonance frequencies of the system [2]. This torque ripple is usually superimposed to the torque ripple of switching frequency. Since the switching frequency has usually high value, its effect will be suppressed by the electrical and mechanical damping of the motor and of the gear.

Fig.3 Closed-fonn analysis results of machine electromagnetic torque with 5% DC-bus voltage pulsation

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In steady state, along with an expected torque ripple of switching frequency, a superimposed pulsation was observed.Particulary, the appearance of the sixth torque harmonic was observed by many researchers [12], [15]. As it was shown in [11] the occurrence of zero vector intervals in PWM influences torque ripple, and leads to a sixth-order torque harmonics. But as it was shown before,

the dc bus voltage pulsation with angular frequency 2 ~ may cause large torque pulsation with the same angular frequency.

\1/ -OJ Tipulst.:

• " , , . , I • , , ,

, I ,

I ,

Fig.4 Closed-fonn analysis results of machine electromagnetic torque pulsation components with 5% DC-bus voltage pulsation

V. EXPERIMENTAL RESULTS

In order to conveniently simulate operation of the voltage source inverter under the condition of pulsating DC-link voltage, an experimental setup shown in Fig. 2 has been built. The DC-link voltage was composed of two separate voltage components. The constant DC voltage was supplied by a three-phase diode rectifier charging a large capacitor bank in the DC bus. The voltage ripple was superimposed to this DC voltage by means of a variable transformer supplied by an auxiliary voltage source inverter (VSI 1) producing a PWM voltage waveform with the fundamental frequency of 100 Hz. Some filtering components were added to the circuit in order to obtain smoother waveforms at the input of the output stage of the main inverter. This output stage generated a six-pulse voltage output feeding an induction machine mechanically loaded by a dynamometer

Proceedings of the 2008 International Conference on Electrical Machines

V~1 ------------------1 ...fn-~_

r-----------------------, , , ,

, , , ~: : -, , ~2~ _________________________________ ~

Fig.5 Experimental setup

From Fig.6 we can see experimental results for the ripple dc-bus voltage 5% and Fig.7 shows results for higher dc-ripple value 10 %.Both Figure show the waveform ofthe dc.bus voltage and corresponding currents in two phases. The phase currents in Fig.6 can be compared with the analytical results shown in Fig.2 (bottom trace ).As one can see the analytical results are in good agreement with the experimental ones.

VI CONCLUSION

Analytical investigation of a three-phase voltage source inverter feeding an induction motor drive under DC-link voltage pulsation is presented in this paper. The analytical expressions for the voltage and current space-vectors as function of the DC-link voltage pulsation were derived. By means of the modified Z-transform and with the mixed p-z mathematical approach we can estimate the separate parts of the solution to evaluate the influence of the DC-link voltage ripple.

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), .. SOOnaA 2..-I ~ 0 11

-:=f- ---------------l---

Fig.6 Experimental results. DC-bus voltage with ripple 5% (top trace).Phase currents (bottom trace)

Fig.7 Experimental results. DC-bus voltage with ripple 10% (top trace).Phase currents (bottom trace)

Proceedings of the 2008 International Conference on Electrical Machines

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[8]. KLIMA J.: Analytical Closed-Form Solution of a Space-Vector Modulated VSI Feeding an Induction Motor Drive.IEEE Transaction on Energy Conversion.VoI.17, No2.June2002, pp.191-196. [9]. KLIMA J.: Mixed p-z Approach for Analytical Analysis of an Induction Motor fed from Space-Vector PWM Voltage Source Inverter. European Trans. on Electric Power (ETEP) Vol.12, NovemberlDecember, 2002. [10] KLIMA J.: Analytical investigation of an induction motor drive under inverter fault conditions. lEE Proc. Electr. Power Appl.Vol.150, 2003 pp.255-262. [II] SLEGER, V., VRECION, P.: MathCAD 7. (In Czech), Haar International, Prague, 1998 [12] DAHONO,P.,SATO.Y.,KATAOKA,T.: Analysis and Minimization of Ripple Components of Output Current and Voltage of PWM Inverters. IEEE Trans. on Industry AppI.VoI.32,No.4,1996,pp.945-9502 [13] STANKOVIC,A.,LIPO,T.: A Novel Control Method for Input-Output Hannonic Elimination of the PWM Boost Type Rectifier Under Unbalanced Operating Conditions. IEEE Trans. on Power Electronics, Vol.16, 2001, pp.603-611. [14] KOLAR,W.,ROUND,S.: Analytical Calculation of the RMS Current Stress on the DC-link Capacitor ofVoltage-PWM Coonverter System- lEE Proc.-Electric Power Appl. Vol. I 53,No.4,July,2006,pp.535-544 [15] CHOMAT,M.,SCHREIER,L.: Control Method for DC-link Voltage Ripple Cancellation in Voltage Source Inverter Under Unbalanced Three-Phase Voltage Supply Conditions. lEE Proc. Electric Power Appl. Vol. I 52 ,Issue 3,May 2005,pp.494-500.

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